Proof of Komlós's conjecture on Hamiltonian subsets

Komlós conjectured in 1981 that among all graphs with minimum degree at least d, the complete graph Kd+1 minimises the number of Hamiltonian subsets, where a subset of vertices is Hamiltonian if it contains a spanning cycle. We prove this conjecture when d is sufficiently large. In fact we prove a stronger result: for large d, any graph G with average degree at least d contains almost twice as many Hamiltonian subsets as Kd+1, unless G is isomorphic to Kd+1 or a certain other graph which we specify.